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Asymptotic distribution and power of the likelihood ratio test for mixtures: bounded and unbounded cases.
 

Summary: Asymptotic distribution and power of the likelihood ratio test for
mixtures: bounded and unbounded cases.
Jean-Marc Aza¨is 1 ´Elisabeth Gassiat2
C´ecile Mercadier1
September 20, 2004
1 Laboratoire de Statistique et Probabilit´es, UMR-CNRS C55830, Universit´e Paul Sabatier,
118 route de Narbonne, 31062 Toulouse Cedex 4. France.
2 Laboratoire de Math´ematiques, Equipe de Probabilit´es, Statistique et Mod´elisation, Bat425,
Universit´e de Paris-Sud, 91405 Orsay Cedex, France. elisabeth.gassiat@math.u-psud.fr.
Abstract
In this paper, we consider the log-likelihood ratio test (LRT) for testing the number of
components in a mixture of populations in a parametric family. We provide the asymptotic
distribution of the LRT statistic under the null hypothesis as well as under contiguous
alternatives when the parameter set is bounded. Moreover, for the simple contamination
model we prove that, under general assumptions, the asymptotic power under contiguous
hypotheses may be arbitrarily close to the asymptotic level when the set of parameters is
large enough. In the particular problem of normal distributions, we prove that, when the
unknown mean is not a priori bounded, the asymptotic power under contiguous hypotheses
is equal to the asymptotic level.
Keywords: Likelihood ratio test, mixture models, number of components, extreme values,

  

Source: Azais, Jean-Marc -Institut de Mathématiques de Toulouse, Université Paul Sabatier
Gassiat, Elisabeth - Département de Mathématiques, Université de Paris-Sud 11

 

Collections: Mathematics